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We have that every holomorphic $\mathbb{CP}^1$-bundle on a compact Riemann surface admits a holomorphic section, due Tsen and as found in Compact Compact Surfaces of Barth, Peters and Van de Ven, for example.

If I add to this bundle a meromorphic flat connection (equivalently a foliation $\mathcal{F}$ which is transversal to the fibers except in finitely many fibers) is it possible to find a transversal section to the foliation $\mathcal{F}$?

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    $\begingroup$ Welcome new contributor. That theorem that you reference is due to Chiungtze C Tsen: en.wikipedia.org/wiki/Chiungtze_C._Tsen It is usually called "Tsen's theorem", although that name is also used for other theorems proved by Tsen. $\endgroup$ – Jason Starr Feb 27 at 11:56
  • $\begingroup$ Thanks, @JasonStarr. I will add this! $\endgroup$ – genas Feb 27 at 13:49
  • $\begingroup$ Consider the special case that the $\mathbb{CP}^1$-bundle over the compact Riemann surface is constant, say $C\times \mathbb{CP}^1$, and the leaf space of the foliation equals the second projection, i.e., the involutive distribution of $\mathcal{F}$ equals the kernel of the derivative $d\text{pr}_2$. Every cross-section is the graph of a morphism $f:C\to \mathbb{CP}^1$. This cross-section is everywhere transversal to $\mathcal{F}$ if and only if $f$ is everywhere submersive. Since $\mathbb{CP}^1$ is simply connected, such $f$ is an isomorphism. $\endgroup$ – Jason Starr Feb 27 at 14:36
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A foliation on a $\mathbb P^1$-bundle over a curve of genus $g$ everywhere transverse to the fibers determines, and is completely determined, by its monodromy representation. The existence of a section everywhere transverse to the foliation is equivalent to endow the curve with a projective structure. Therefore, in the particular case of foliations everywhere transverse to the fibration you are asking for the possible monodromy representations of projective structures on it.

This problem has been completely solved by Gallo-Kapovitch-Marden. For a survey on this circle of ideas, see this paper by Loray-Marin.

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I am posting my comment above as an answer. There are two different questions: does there exist a section that is generically transversal to the foliation, and does there exist a section that is everywhere transversal to the foliation.

The second question has a negative answer. Indeed, for a compact Riemann surface $C$ of positive genus, consider the product $\mathbb{CP}^1$-bundle, $C\times \mathbb{CP}^1$. For the second projection, $$\text{pr}_2:C\times \mathbb{CP}^1,$$ the fibers of $\text{pr}_2$ form the leaves of an everywhere regular foliation $\mathcal{F}$ on $C\times \mathbb{CP}^1$. Every cross-section of $\text{pr}_1$ is the graph of a morphism, $$f:C\to \mathbb{CP}^1.$$ This cross-section is transversal to $\mathcal{F}$ at precisely the image of the submersive locus of $f$. An everywhere submersive holomorphic map between compact, connected Riemann surfaces is an unbranched cover. Since $\mathbb{CP}^1$ is simply connected, there is no such unbranched cover. Thus, there is no cross-section that is everywhere transversal to $\mathcal{F}$.

On the other hand, there always exist sections that are generically transversal to $\mathcal{F}$. Indeed, one consequence of Tsen's theorem is that every $\mathbb{CP}^1$-bundle is bimeromorphic to a product family. Every (saturated) meromorphic foliation on the product family is a regular holomorphic foliation away from finitely many fibers. For a regular fiber $\{x\}\times \mathbb{CP}^1$, for every point $t\in \mathbb{CP}^1$, the fiber of the foliation at this point is a $1$-dimensional subspace $L$ of the $2$-dimensional vector space $T_{C,x}\oplus T_{\mathbb{CP}^1,t}$.

By Riemann-Roch, etc., there exists a nonconstant morphism from $C$ to $\mathbb{CP}^1$ that is submersive at $x$. Since the automorphism group of $\mathbb{CP}^1$ is transitive, we can post-compose to arrange that $x$ maps to $t$. Finally, since the automorphism group of $(\mathbb{CP}^1,x)$ acts transitively on $T_{\mathbb{CP}^1,x}\setminus\{0\}$, we can post-compose further and arrange that the graph of the morphism $$f:(C,x)\to (\mathbb{CP}^1,t),$$ at $(x,t)$ has tangent space that is different from $L$. Thus, the graph of $f$ gives a cross-section that is transversal at one point. The transversal locus is Zariski open, hence the transversal locus is the section.

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